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并查集</h1><div id="post-meta"><div class="meta-firstline"><span class="post-meta-date"><i class="far fa-calendar-alt fa-fw post-meta-icon"></i><span class="post-meta-label">发表于</span><time class="post-meta-date-created" datetime="2021-12-06T09:36:00.000Z" title="发表于 2021-12-06 17:36:00">2021-12-06</time><span class="post-meta-separator">|</span><i class="fas fa-history fa-fw post-meta-icon"></i><span class="post-meta-label">更新于</span><time class="post-meta-date-updated" datetime="2021-12-10T12:59:46.903Z" title="更新于 2021-12-10 20:59:46">2021-12-10</time></span><span class="post-meta-categories"><span class="post-meta-separator">|</span><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/hexo3/categories/%E7%AE%97%E6%B3%95%E5%9F%BA%E7%A1%80/">算法基础</a></span></div><div class="meta-secondline"><span class="post-meta-separator">|</span><span class="post-meta-wordcount"><i class="far fa-file-word fa-fw post-meta-icon"></i><span class="post-meta-label">字数总计:</span><span class="word-count">820</span><span class="post-meta-separator">|</span><i class="far fa-clock fa-fw post-meta-icon"></i><span class="post-meta-label">阅读时长:</span><span>3分钟</span></span><span class="post-meta-separator">|</span><span class="post-meta-pv-cv" id="" data-flag-title="算法 - 并查集"><i class="far fa-eye fa-fw post-meta-icon"></i><span class="post-meta-label">阅读量:</span><span id="busuanzi_value_page_pv"></span></span></div></div></div></header><main class="layout" id="content-inner"><div id="post"><article class="post-content" id="article-container"><h1 id="算法-并查集"><a href="#算法-并查集" class="headerlink" title="算法 - 并查集"></a>算法 - 并查集</h1><!-- GFM-TOC -->
<ul>
<li><a href="#%E7%AE%97%E6%B3%95---%E5%B9%B6%E6%9F%A5%E9%9B%86">算法 - 并查集</a><ul>
<li><a href="#%E5%89%8D%E8%A8%80">前言</a></li>
<li><a href="#quick-find">Quick Find</a></li>
<li><a href="#quick-union">Quick Union</a></li>
<li><a href="#%E5%8A%A0%E6%9D%83-quick-union">加权 Quick Union</a></li>
<li><a href="#%E8%B7%AF%E5%BE%84%E5%8E%8B%E7%BC%A9%E7%9A%84%E5%8A%A0%E6%9D%83-quick-union">路径压缩的加权 Quick Union</a></li>
<li><a href="#%E6%AF%94%E8%BE%83">比较</a><!-- GFM-TOC --></li>
</ul>
</li>
</ul>
<h2 id="前言"><a href="#前言" class="headerlink" title="前言"></a>前言</h2><p>用于解决动态连通性问题，能动态连接两个点，并且判断两个点是否连通。</p>
<div align="center"> <img src= "" data-lazy-src="https://cs-notes-1256109796.cos.ap-guangzhou.myqcloud.com/02943a90-7dd4-4e9a-9325-f8217d3cc54d.jpg" width="350"/> </div><br>

<table>
<thead>
<tr>
<th align="center">方法</th>
<th align="center">描述</th>
</tr>
</thead>
<tbody><tr>
<td align="center">UF(int N)</td>
<td align="center">构造一个大小为 N 的并查集</td>
</tr>
<tr>
<td align="center">void union(int p, int q)</td>
<td align="center">连接 p 和 q 节点</td>
</tr>
<tr>
<td align="center">int find(int p)</td>
<td align="center">查找 p 所在的连通分量编号</td>
</tr>
<tr>
<td align="center">boolean connected(int p, int q)</td>
<td align="center">判断 p 和 q 节点是否连通</td>
</tr>
</tbody></table>
<figure class="highlight java"><table><tr><td class="code"><pre><span class="line"><span class="keyword">public</span> <span class="keyword">abstract</span> <span class="class"><span class="keyword">class</span> <span class="title">UF</span> </span>&#123;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">protected</span> <span class="keyword">int</span>[] id;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="title">UF</span><span class="params">(<span class="keyword">int</span> N)</span> </span>&#123;</span><br><span class="line">        id = <span class="keyword">new</span> <span class="keyword">int</span>[N];</span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i &lt; N; i++) &#123;</span><br><span class="line">            id[i] = i;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">boolean</span> <span class="title">connected</span><span class="params">(<span class="keyword">int</span> p, <span class="keyword">int</span> q)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">return</span> find(p) == find(q);</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">abstract</span> <span class="keyword">int</span> <span class="title">find</span><span class="params">(<span class="keyword">int</span> p)</span></span>;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">abstract</span> <span class="keyword">void</span> <span class="title">union</span><span class="params">(<span class="keyword">int</span> p, <span class="keyword">int</span> q)</span></span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<h2 id="Quick-Find"><a href="#Quick-Find" class="headerlink" title="Quick Find"></a>Quick Find</h2><p>可以快速进行 find 操作，也就是可以快速判断两个节点是否连通。</p>
<p>需要保证同一连通分量的所有节点的 id 值相等，就可以通过判断两个节点的 id 值是否相等从而判断其连通性。</p>
<p>但是 union 操作代价却很高，需要将其中一个连通分量中的所有节点 id 值都修改为另一个节点的 id 值。</p>
<div align="center"> <img src= "" data-lazy-src="https://cs-notes-1256109796.cos.ap-guangzhou.myqcloud.com/0972501d-f854-4d26-8fce-babb27c267f6.jpg" width="320"/> </div><br>

<figure class="highlight java"><table><tr><td class="code"><pre><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">QuickFindUF</span> <span class="keyword">extends</span> <span class="title">UF</span> </span>&#123;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="title">QuickFindUF</span><span class="params">(<span class="keyword">int</span> N)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">super</span>(N);</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">    <span class="meta">@Override</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">int</span> <span class="title">find</span><span class="params">(<span class="keyword">int</span> p)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">return</span> id[p];</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">    <span class="meta">@Override</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">void</span> <span class="title">union</span><span class="params">(<span class="keyword">int</span> p, <span class="keyword">int</span> q)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">int</span> pID = find(p);</span><br><span class="line">        <span class="keyword">int</span> qID = find(q);</span><br><span class="line"></span><br><span class="line">        <span class="keyword">if</span> (pID == qID) &#123;</span><br><span class="line">            <span class="keyword">return</span>;</span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i &lt; id.length; i++) &#123;</span><br><span class="line">            <span class="keyword">if</span> (id[i] == pID) &#123;</span><br><span class="line">                id[i] = qID;</span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<h2 id="Quick-Union"><a href="#Quick-Union" class="headerlink" title="Quick Union"></a>Quick Union</h2><p>可以快速进行 union 操作，只需要修改一个节点的 id 值即可。</p>
<p>但是 find 操作开销很大，因为同一个连通分量的节点 id 值不同，id 值只是用来指向另一个节点。因此需要一直向上查找操作，直到找到最上层的节点。</p>
<div align="center"> <img src= "" data-lazy-src="https://cs-notes-1256109796.cos.ap-guangzhou.myqcloud.com/11b27de5-5a9d-45e4-95cc-417fa3ad1d38.jpg" width="280"/> </div><br>

<figure class="highlight java"><table><tr><td class="code"><pre><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">QuickUnionUF</span> <span class="keyword">extends</span> <span class="title">UF</span> </span>&#123;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="title">QuickUnionUF</span><span class="params">(<span class="keyword">int</span> N)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">super</span>(N);</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">    <span class="meta">@Override</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">int</span> <span class="title">find</span><span class="params">(<span class="keyword">int</span> p)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">while</span> (p != id[p]) &#123;</span><br><span class="line">            p = id[p];</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">return</span> p;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">    <span class="meta">@Override</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">void</span> <span class="title">union</span><span class="params">(<span class="keyword">int</span> p, <span class="keyword">int</span> q)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">int</span> pRoot = find(p);</span><br><span class="line">        <span class="keyword">int</span> qRoot = find(q);</span><br><span class="line"></span><br><span class="line">        <span class="keyword">if</span> (pRoot != qRoot) &#123;</span><br><span class="line">            id[pRoot] = qRoot;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<p>这种方法可以快速进行 union 操作，但是 find 操作和树高成正比，最坏的情况下树的高度为节点的数目。</p>
<div align="center"> <img src= "" data-lazy-src="https://cs-notes-1256109796.cos.ap-guangzhou.myqcloud.com/23e4462b-263f-4d15-8805-529e0ca7a4d1.jpg" width="100"/> </div><br>

<h2 id="加权-Quick-Union"><a href="#加权-Quick-Union" class="headerlink" title="加权 Quick Union"></a>加权 Quick Union</h2><p>为了解决 quick-union 的树通常会很高的问题，加权 quick-union 在 union 操作时会让较小的树连接较大的树上面。</p>
<p>理论研究证明，加权 quick-union 算法构造的树深度最多不超过 logN。</p>
<div align="center"> <img src= "" data-lazy-src="https://cs-notes-1256109796.cos.ap-guangzhou.myqcloud.com/a9f18f8a-c1ea-422e-aa56-d91716b0f755.jpg" width="150"/> </div><br>

<figure class="highlight java"><table><tr><td class="code"><pre><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">WeightedQuickUnionUF</span> <span class="keyword">extends</span> <span class="title">UF</span> </span>&#123;</span><br><span class="line"></span><br><span class="line">    <span class="comment">// 保存节点的数量信息</span></span><br><span class="line">    <span class="keyword">private</span> <span class="keyword">int</span>[] sz;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="title">WeightedQuickUnionUF</span><span class="params">(<span class="keyword">int</span> N)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">super</span>(N);</span><br><span class="line">        <span class="keyword">this</span>.sz = <span class="keyword">new</span> <span class="keyword">int</span>[N];</span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i &lt; N; i++) &#123;</span><br><span class="line">            <span class="keyword">this</span>.sz[i] = <span class="number">1</span>;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">    <span class="meta">@Override</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">int</span> <span class="title">find</span><span class="params">(<span class="keyword">int</span> p)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">while</span> (p != id[p]) &#123;</span><br><span class="line">            p = id[p];</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">return</span> p;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">    <span class="meta">@Override</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">void</span> <span class="title">union</span><span class="params">(<span class="keyword">int</span> p, <span class="keyword">int</span> q)</span> </span>&#123;</span><br><span class="line"></span><br><span class="line">        <span class="keyword">int</span> i = find(p);</span><br><span class="line">        <span class="keyword">int</span> j = find(q);</span><br><span class="line"></span><br><span class="line">        <span class="keyword">if</span> (i == j) <span class="keyword">return</span>;</span><br><span class="line"></span><br><span class="line">        <span class="keyword">if</span> (sz[i] &lt; sz[j]) &#123;</span><br><span class="line">            id[i] = j;</span><br><span class="line">            sz[j] += sz[i];</span><br><span class="line">        &#125; <span class="keyword">else</span> &#123;</span><br><span class="line">            id[j] = i;</span><br><span class="line">            sz[i] += sz[j];</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<h2 id="路径压缩的加权-Quick-Union"><a href="#路径压缩的加权-Quick-Union" class="headerlink" title="路径压缩的加权 Quick Union"></a>路径压缩的加权 Quick Union</h2><p>在检查节点的同时将它们直接链接到根节点，只需要在 find 中添加一个循环即可。</p>
<h2 id="比较"><a href="#比较" class="headerlink" title="比较"></a>比较</h2><table>
<thead>
<tr>
<th align="center">算法</th>
<th align="center">union</th>
<th align="center">find</th>
</tr>
</thead>
<tbody><tr>
<td align="center">Quick Find</td>
<td align="center">N</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">Quick Union</td>
<td align="center">树高</td>
<td align="center">树高</td>
</tr>
<tr>
<td align="center">加权 Quick Union</td>
<td align="center">logN</td>
<td align="center">logN</td>
</tr>
<tr>
<td align="center">路径压缩的加权 Quick Union</td>
<td align="center">非常接近 1</td>
<td align="center">非常接近 1</td>
</tr>
</tbody></table>
</article><div class="post-copyright"><div class="post-copyright__author"><span class="post-copyright-meta">文章作者: </span><span class="post-copyright-info"><a href="mailto:undefined">Zhang Shuo</a></span></div><div class="post-copyright__type"><span class="post-copyright-meta">文章链接: </span><span class="post-copyright-info"><a href="https://zhang-shuo-fr.gitee.io/hexo3/2021/12/06/notes/%E7%AE%97%E6%B3%95%20-%20%E5%B9%B6%E6%9F%A5%E9%9B%86/">https://zhang-shuo-fr.gitee.io/hexo3/2021/12/06/notes/%E7%AE%97%E6%B3%95%20-%20%E5%B9%B6%E6%9F%A5%E9%9B%86/</a></span></div><div class="post-copyright__notice"><span class="post-copyright-meta">版权声明: </span><span class="post-copyright-info">本博客所有文章除特别声明外，均采用 <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" target="_blank">CC BY-NC-SA 4.0</a> 许可协议。转载请注明来自 <a href="https://zhang-shuo-fr.gitee.io/hexo3" target="_blank">Zhang Shuo'blog</a>！</span></div></div><div class="tag_share"><div class="post-meta__tag-list"><a class="post-meta__tags" href="/hexo3/tags/%E7%AE%97%E6%B3%95%E5%9F%BA%E7%A1%80/">算法基础</a></div><div class="post_share"><div class="social-share" data-image="/hexo3/img/19.jpg" data-sites="facebook,twitter,wechat,weibo,qq"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/social-share.js/dist/css/share.min.css" media="print" onload="this.media='all'"><script src="https://cdn.jsdelivr.net/npm/social-share.js/dist/js/social-share.min.js" defer></script></div></div><div class="post-reward"><div class="reward-button button--animated"><i class="fas fa-qrcode"></i> 打赏</div><div class="reward-main"><ul class="reward-all"><li class="reward-item"><a href="/hexo3/img/wechat.jpg" target="_blank"><img class="post-qr-code-img" src= "" data-lazy-src="/hexo3/img/wechat.jpg" alt="微信"/></a><div class="post-qr-code-desc">微信</div></li><li class="reward-item"><a href="/hexo3/img/alipay.jpg" target="_blank"><img class="post-qr-code-img" src= "" data-lazy-src="/hexo3/img/alipay.jpg" alt="支付宝"/></a><div class="post-qr-code-desc">支付宝</div></li></ul></div></div><nav class="pagination-post" id="pagination"><div class="prev-post pull-left"><a href="/hexo3/2021/12/06/notes/%E7%AE%97%E6%B3%95%20-%20%E6%A0%88%E5%92%8C%E9%98%9F%E5%88%97/"><img class="prev-cover" src= "" data-lazy-src="/hexo3/img/11.jpg" onerror="onerror=null;src='/hexo3/img/404.jpg'" alt="cover of previous post"><div class="pagination-info"><div class="label">上一篇</div><div class="prev_info">算法 - 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排序"><img class="cover" src= "" data-lazy-src="/hexo3/img/9.jpg" alt="cover"><div class="content is-center"><div class="date"><i class="far fa-calendar-alt fa-fw"></i> 2021-12-06</div><div class="title">算法 - 排序</div></div></a></div></div></div></div><div class="aside-content" id="aside-content"><div class="sticky_layout"><div class="card-widget" id="card-toc"><div class="item-headline"><i class="fas fa-stream"></i><span>目录</span></div><div class="toc-content"><ol class="toc"><li class="toc-item toc-level-1"><a class="toc-link" href="#%E7%AE%97%E6%B3%95-%E5%B9%B6%E6%9F%A5%E9%9B%86"><span class="toc-number">1.</span> <span class="toc-text">算法 - 并查集</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%89%8D%E8%A8%80"><span class="toc-number">1.1.</span> <span class="toc-text">前言</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#Quick-Find"><span class="toc-number">1.2.</span> <span class="toc-text">Quick Find</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#Quick-Union"><span class="toc-number">1.3.</span> <span class="toc-text">Quick Union</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%8A%A0%E6%9D%83-Quick-Union"><span class="toc-number">1.4.</span> <span class="toc-text">加权 Quick Union</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E8%B7%AF%E5%BE%84%E5%8E%8B%E7%BC%A9%E7%9A%84%E5%8A%A0%E6%9D%83-Quick-Union"><span class="toc-number">1.5.</span> <span class="toc-text">路径压缩的加权 Quick Union</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%AF%94%E8%BE%83"><span class="toc-number">1.6.</span> <span class="toc-text">比较</span></a></li></ol></li></ol></div></div></div></div></main><footer id="footer" style="background-image: url('/hexo3/img/19.jpg')"><div id="footer-wrap"><div class="copyright">&copy;2020 - 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